Basel II Second Pillar: an Analytical VaR with Contagion and Sectorial Risks

Basel II Second Pillar: an Analytical VaR with Contagion and Sectorial Risks Michele Bonollo Paola Mosconi Fabio Mercurio January 29, 2009 Abstract This paper deals with the effects of concentration (single name and sectoral) and contagion risk on credit portfolios. Results are obtained for the value at risk of the portfolio loss distribution, in the analytical framework originally developed by Vasicek in 1991 [1]. V ar is expressed as a sum of terms: the first contribution represents the value at risk of a hypothetical single-factor homogeneous portfolio, the remaining terms are corrections due to contagion, imperfect granularity and multiple industry-geographic sectors. A detailed numerical analysis is also presented. 1 Introduction Concentration and contagion risk on credit portfolios have been studied for many years with different methodologies and approaches. Such risks can be seen as departures from the Asymptotic Single-Risk Factor (ASRF) paradigm which underlies the IRB approaches of Basel II [2]. Basic hypothesis of this model include the homogeneity of the underlying portfolio and a common factor driving systematic risk. In this framework, concentration risk represents a violation of the ASFR model and can be decomposed into two contributions: an idiosyncratic part, single name or imperfect granularity risk, due to the small size of the portfolio or to the presence of large exposures associated to single obligors and a systematic term, sectoral concentration, due to imperfect diversification across sectoral factors. Many portfolio models have been developed in order to deal with concentration risk (e.g. CreditMetrics [3], PortfolioManager [4], CreditPortfolioView [5] etc...) and some of them rely Banco Popolare and Università di Padova - michele.bonollo@sgsbp.it Iason Ltd - paola.mosconi@iasonltd.com Bloomberg and Iason Ltd 1

on computationally heavy Monte Carlo simulations. A different solution to the problem of calculating economic capital exploits an approximated analytical technique which applies to one-factor Merton type models. This method, originally introduced by Vasicek [1], consists in replacing the original portfolio loss distribution with an asymptotic one, whose value at risk (V ar) can be computed analytically. The difference between the true and the asymptotic V ar can also be computed analytically through a second order approximation [6]. Many steps have been taken in this direction, extending the original Vasicek result for homogeneous portfolios to include granularity risk [7], [8], [9], [10] and sectoral concentration risk (see Pykhtin [11]). The third source of risk, credit contagion lies somewhat in-between the previous two [2]. This risk takes into account the occurrence of default events triggered by inter-dependencies (legal, financial, business-oriented) among obligors. Very diverse approaches have been proposed to tackle this problem. Davis and Lo [12] have built a first model where the default of any company in the portfolio can infect all the others. Egloff et al [13] have developed a neural-network inspired model to mimic the structure of links among obligors in a portfolio. Recently Rösch et al [14] have proposed an extension of [12] in a default mode scenario where obligors are divided into two categories: those who can be considered immune from contagion I-firms (infecting) and those who can be contaminated C-firms. In this paper, we study in a unified framework the effects of concentration and contagion risk. A first attempt to generalize the work by Pykthin in this direction has been pursued by [15]. However, the resulting model specification appears incomplete to some extent and hardly applicable to concrete problems. Here we combine this idea with the contagion specification proposed by Rösch [14], in order to obtain a model which is general enough still preserving analytical tractability. The paper is organized as follows. We first introduce the model in Section 2, including a detailed specification of the contagion part. We present the core idea of V ar decomposition and the corresponding analytical results in Section 3 and a detailed numerical analysis in Section 4. Section 5 summarizes and collects our final remarks. Technical details and issues can be found in the Appendix. 2 The model The starting point is a multi-factor default mode Merton model. The portfolio is assumed to have the following features: loans are associated to M distinct borrowers. Each borrower has exactly one loan characterized by exposure EAD i. We define the weight of a loan in the portfolio as w i = EAD i / M i=1 EAD i Each borrower has default probability p i within a chosen time horizon. If borrower i defaults, the amount of loss is determined by its loss-given default, LGD i. Such quantity is assumed to be stochastic with mean µ i and standard deviation σ i and to be independent of the other LGDs and stochastic variables present in the model. 2

First we introduce the multi-factor set up, without considering contagion, along the lines traced by Pykhtin [11]. Subsequently, we extend the model in order to include the effects of contagion risk. 2.1 Multi-factor model specification Asset returns {X i } i=1,...,m are assumed to be distributed according to a standard normal distribution and to be linearly dependent on N normally distributed systematic risk factors. Expressing the contribution coming from these sectoral factors in terms of a composite variable {Y i } i=1,...,m, one for each borrower, the behavior of the ith borrower s asset return is X i = r i Y i + 1 ri 2ξ i. (1) r i is the sensitivity of borrower i to the systematic risk, corr(x i, X j ) = ρ i,j and ξ i N(0, 1), i.i.d, represents the idiosyncratic component of risk which can be diversified away in the case of an infinitely granular portfolio. The composite factor can be expressed as a linear combination of N independent systematic factors Z k N(0, 1), k = 1,...,N Y i = N α ik Z k, (2) k=1 and it is assumed to have unit variance, i.e. N returns can be cast into the general form X i = a i Y + k=1 α2 ik N (r i α ik a i b k )Z k + k=1 This formula can be analyzed as follows: the last term is the idiosyncratic component already discussed. = 1. It turns out that asset 1 r 2 i ξ i. (3) The first two terms account for the systematic (sectoral) risk contribution. Y = N k=1 b kz k is an effective systematic factor, characterized by unit variance i.e. N k=1 b2 k = 1. The coefficients {a i} are effective factor loadings, given by a i = r N i k=1 α ikb k. They can be derived through an optimization procedure, as shown in [11]. The second term N k=1 (r iα ik a i b k )Z k is independent of Y and encodes the conditional asset correlation ρ Y ij = r N ir j k=1 α ikα jk a i a j (1. (4) a 2 i )(1 a2 j ) 3

2.2 Credit contagion Contagion risk can be ascribed to inter-company ties, such as legal (parent-subsidiary) relationships, financial and business oriented relations (supplier-purchaser interactions) and so on. This entails a complex network of links among obligors, which makes the credit contagion problem very hard to solve. Here we adopt a simplified perspective. We assume that obligors are broadly divided into two categories: those firms which are immune from contagion (referred to as I-firms, i.e. infecting) and those companies which can be contaminated by the first group through credit contagion ( C-firms ). Asset returns associated to group I follow the multi-factor specification given by eq. (1) while C-firms asset returns satisfy X i = r i Y i + 1 ri 2 ξ(γ i, ǫ i ). (5) The firm-specific factor ξ(γ i, ǫ i ) can be expressed as ξ(γ i, ǫ i ) = g i Γ i + 1 gi 2 ǫ i (6) where ǫ i is the usual idiosyncratic contribution, the term g i Γ i encodes the effects of contagion risk. The composite contagion factor Γ i can be written as a sum over latent contagion variables C l (assumed to be independent and distributed as N(0, 1)) Γ i = N γ il C l. l=1 The unit variance property of X i is preserved if N l=1 γ2 il = 1. We assume that companies can be mapped into the industry-geographic sector with which they have the highest correlation and we decompose each sector into a I segment and a C one. Therefore, the contagion effect experienced by an arbitrary Cfirm can be thought of as the weighted sum of contributions coming from the infecting segments of different sectors. Under this specification, the number of latent contagion factors equals the number of industry-geographic factors, N. The coefficient g i plays the role of a contagion factor loading and represents a measure of how much obligor i is overall affected by contagion. It is worth noticing that eq.s (5-6) express in compact form also the behavior of I-firms, with the understanding that g i = 0 in that case. We will come back to the estimation of the contagion parameters in the Appendix. Having specified the model in this way, by making explicit the effects due to multi- 4

factors and contagion, asset returns turn out to follow X i = a i Y + + + N (r i α ik a i b k )Z k + k=1 1 r 2 i g i 1 r 2 i N γ il C l + (7) l=1 1 g 2 i ǫ i. The conditional correlation between distinct obligors i and j assumes the form r i r N j ρ Y ij +C k=1 α ikα jk + 1 ri 2 1 rj 2 g K ig j l=1 γ ilγ jl a i a j =. (8) (1 a 2 i )(1 a2 j ) 3 V ar decomposition and results Given this setup, the portfolio loss rate L can be written as L = M w i L i = i=1 M w i 1 {Xi N 1 (p i )} LGD i. (9) i=1 Our goal is to calculate the quantile at confidence level q of this quantity, namely t q (L). It has been shown ([8] and [6]) that such a task can be accomplished through a Taylor expansion around the quantile of another variable L, such that t q (L) is analytically tractable and sufficiently close to t q (L). The results in the multi-factor case and in the model with contagion are collected in the next two subsections. 3.1 Multi-factor model Let the variable L be defined as the limiting loss distribution in the one-factor Merton framework M L = l(y ) = w i µ i ˆp i (Y ), (10) i=1 where ˆp i (y) is the probability of default of borrower i, conditional on Y = y ˆp i (y) = N N 1 (p i ) a i y. (11) 1 a 2 i (N indicates the cumulative normal distribution). The quantile of L at level q can be calculated analytically as t q (L) = l(n 1 (1 q)). 5

Carrying out the Taylor expansion of t q (L) around t q (L), first-order contributions cancel out and the final result, up to the second order, assumes the form [11] t q t q (L) t q (L) = 1 2 l (y) [ ( l ν )] (y) (y) ν(y) l (y) + y y=n 1 (1 q) The function l(y) is defined as in (10), while ν(y) = var[l Y = y] is the conditional variance of L on Y = y. This function can be further decomposed in terms of its systematic and idiosyncratic components where (12) ν(y) = ν (y) + ν GA (y), (13) ν (y) = var[e(l {Z k }) Y = y] = (14) M M [ = w i w j µ i µ j N2 (N 1 [ˆp i (y)], N 1 [ˆp j (y)], ρ Y ij) ˆp i (y)ˆp j (y) ], i=1 j=1 ν GA (y) = E[var(L {Z k }) Y = y] = (15) M ( = µ 2 i [ˆpi (y) N 2 (N 1 [ˆp i (y)], N 1 [ˆp i (y)], ρ Y ii) ] + σi 2 ˆp i (y) ). i=1 w 2 i (N 2 (.,.,.) is the bivariate normal cumulative distribution function). The first term, ν (y), accounts for the correction to the loss distribution due to the multi-factor setting, in the limit of an infinitely fine-grained portfolio. ν GA (y) is the granularity adjustment term. The quantity ρ Y ii is obtained by replacing the index j with i in eq. (4). In the special case of homogeneous LGDs and default probabilities p i, it becomes proportional to the Herfindahl-Hirschman index HHI = M i=1 w2 i (see [9]). Explicit expressions for the derivatives of l(y) and ν(y) can be found in the Appendix. 3.2 Contagion effects Repeating the analysis performed in the previous paragraph, the results obtained for the multi-factor set up can be easily extended in order to include contagion risk. Eq. (12) is still valid, with the understanding that now the conditional variance ν(y) must be replaced by where ν C (y) = ν C (y) + ν C GA(y), ν (y) C = var[e(l {Z k, C l }) Y = y] = (16) M M [ ] = w i w j µ i µ j N 2 (N 1 [ˆp i (y)], N 1 [ˆp j (y)], ρ Y +C ) ˆp i (y)ˆp j (y), i=1 j=1 6 ij

νga(y) C = E[var(L {Z k, C l }) Y = y] = (17) M ( [ ] ) = wi 2 µ 2 i ˆp i (y) N 2 (N 1 [ˆp i (y)], N 1 [ˆp i (y)], ρ Y ii +C ) + σi 2 ˆp i (y). i=1 In this case, ν C encodes the correction to the loss distribution due to multi-factor and contagion effects, in the limit of an infinitely granular portfolio. ν C GA represent the granularity contribution. The conditional correlation appearing in eq.s (16-17) is now given by formula (8). 4 Applications This section is devoted to applications of the theoretical model and numerical examples. In the first subsection we show and comment upon the results we get, highlighting the role played by the numerous parameters which specify the model. In the second part, we compare the values of the value at risk we obtain with those derived through Monte Carlo simulations by Gordy (2000) [16] and Carey (2001) [17]. As it will be evident, the agreement is very satisfactory. 4.1 Numerical analysis We aim at showing how the value at risk calculated through the approximated formula at the second order, eq. (12), behaves as a function of the parameters which define the model. In particular, we focus on the number of obligors M, the number of systematic risk factors N (which also affects the contagion specification) and the rating quality of the portfolio. However, before entering the details, some general information about the characterization of the portfolio and the model itself is needed. Portfolio data and features of the model Loan exposures are assigned following the rule EAD i = i 3 [9]. For M 100 and above, the loan to one borrower limit of 4% of the total portfolio size is not exceeded. We sort obligors in ascending order with respect to their exposure. We further assume that the last 20% of them belongs to the group of infecting I-firms. In the following, we assume for sake of simplicity that industry-geographical sectors are already expressed in terms of independent standardized normal risk factors (the {Z k } variables of the theoretical model) 1. The information about the dependence of single obligors on these factors is encoded into obligor/sector correlation coefficients, i.e. ρ ik = r i α ik, where i 1 The most general case of dependent sectors can be reduced to this simplified scenario by means of a Cholesky decomposition of the variance-covariance matrix or through the principal component analysis (PCA). 7

represents the i-th obligor and k the sector. Assuming that banks have these kind of data at their disposal, we further simplify the model by introducing classes of correlation. For instance, if we only considered three classes (low, average and high correlation), we would assign all correlations below 33% the value ρ ik = 16.5%, those between 34% and 66%, ρ ik = 49.5% and so on. Given the coefficients ρ ik, the factor loadings r i can be easily deduced by observing that r 2 i = N k=1 ρ2 ik. In an analogous way, α ik can be derived from ρ ik = r i α ik. (In the following analysis we consider five correlation classes.) Very similar observations can be applied to contagion factors. For simplicity, also latent variables C l are assumed to be N(0, 1) i.i.d. The coefficients γ ik, which encode the dependence of single obligors on the infecting segments of each industry-geographic area, are grouped into classes as well. However, they satisfy a different constraint, N k=1 γ2 ik = 1. (Also in this case, we opt for five contagion classes.) In the contagion specification, we are left with another parameter, to be chosen at each bank s discretion: the factor loading g i. Similarly, we group its possible values into five classes. As far as the quality of the portfolio is concerned, we consider three different possibilities, based on a seven rating classes subdivision (see Gordy 2000 [16] for details): 1. high quality portfolio, characterized by a percentage of speculative grade loans (BB and below) less than 25%, 2. average quality portfolio, such that speculative grade loans account for 50% of the total exposure, 3. low quality portfolio, made up of roughly 79% of speculative grade loans. This can be visualized in Fig. 1: Eventually, following the specification introduced by Gordy 2000 [16], we choose a constant LGD mean value, across different rating classes, given by µ i = 30%. The corresponding standard deviation turns out to be σ i = 1/2 µ i (1 µ i ) 0.229. We now address the core discussion of the numerical analysis. V ar vs number of obligors We assume an average quality portfolio, characterized by N = 5 industry-geographic sectors, and, correspondingly, five latent contagion risk factors. We fix the confidence level at q = 99.5% and calculate the value at risk for M, the number of obligors, ranging from 100 to 1000. We store the results in Table 1: The first column collects the Hirschmann-Herfindahl index (HHI) for each scenario, giving an idea of the granularity of the underlying portfolios. As it appears clearly, upon increasing the number of loans, M, such a contribution becomes less relevant. 8

Quality and rating 3 2 1 AAA AA A BBB BB B CCC 0% 20% 40% 60% 80% 100% Figure 1: Subdivision into seven rating classes for a high (1), average (2) and poor (3) quality portfolio. HHI V ar 99.5% Y C C GA cpu-time M=100 0.0227 0.0386 0.0175 0.0014 0.0151 0.0024 41.0190 M=200 0.0114 0.0288 0.0112 0.0022 0.0079 0.0034 162.8141 M=500 0.0046 0.0252 0.0059 0.0020 0.0030 0.0029 1020.0000 M=1000 0.0023 0.0225 0.0048 0.0024 0.0015 0.0033 4100.0000 Table 1: Results obtained for an average quality portfolio, characterized by seven rating classes, five industry-geographic areas and contagion factors, at the level of confidence q = 99.5%. The following columns contain respectively the value at risk of the loss distribution L, given by the approximated formula (12) in the presence of sector and contagion risk, and the correction to the homogeneous single-factor asymptotic V ar, decomposed into its main contributions: Y C = total correction due to multi-factor and contagion; C = total correction due to contagion; GA = granularity adjustment; = correction due to multi-sector and contagion, for a homogeneous portfolio. Results from columns (2) and (6) are represented graphically in Fig. 2 As it emerges from the picture, the value of V ar 99.5% t 99.5% (L) decreases of about 160 basis points, moving from 100 obligors to 1000. This behavior is consistent with the fact that the portfolio becomes more homogeneous as the number of loans increases, leading to a diminishing granularity adjustment. The corrections due to the multi-sector set up and the effects of contagion are only mildly affected by the number of obligors. 9

Approximated VaR (Sector + contagion risk) 0.04 0.016 Granularity adjustment 0.038 0.014 0.036 0.034 0.012 0.01 VaR 99.5% 0.032 0.03 GA 0.008 0.028 0.006 0.026 0.004 0.024 0.002 0.022 100 400 700 1000 Number of Obligors M 0 200 400 600 800 1000 Number of Obligors M Figure 2: Approximated V ar 99.5% and granularity adjustment as a function of M (number of obligors), for an average quality portfolio, characterized by seven rating classes, five industry-geographic sectors and contagion factors. V ar vs number of sectors We now let the number of systematic risk factors vary from 1 to 10. The results for an average quality portfolio, characterized by M = 300 obligors (HHI = 0.0076) and level of confidence q = 99.5% are summarized in Table 2: t q(l) V ar 99.5% Y C C GA N=1 0.0370 0.0442 0.0072 0.0035 0.0036 0.0037 N=2 0.0300 0.0371 0.0071 0.0029 0.0038 0.0033 N=6 0.0153 0.0238 0.0085 0.0016 0.0060 0.0025 N=10 0.0116 0.0226 0.0110 0.0018 0.0081 0.0029 Table 2: Results for and average quality portfolio, characterized by seven rating classes, M = 300 obligors (HHI = 0.0076) and level of confidence q = 99.5%. The first column displays the data corresponding to the homogeneous, singlefactor V ar, t q (L). As it appears clearly, upon increasing the number of sectors, diversification benefits turns out to play a central role. This effect is particularly evident in the asymptotic component t q (L), thanks to the effective factor loading a i, which takes into account the combined contribution of all sectors (see Section 2). The result is shown in Fig. 3. We eventually notice that the correction, directly related to the multi-sector set up, assumes almost constant values. This is consistent with our choice, at the level of simulation, of picking N sectors globally, but requiring each obligor to interact with at most two industry-geographic areas simultaneously. Similarly, we have chosen each C-firm to be infected by at most three infecting segments beyond the one in its own 10

0.045 Approximated VaR (Sector + contagion risk ) 0.04 0.035 VaR 99.5% 0.03 0.025 0.02 1 2 3 4 5 6 7 8 9 10 Number of sectors N Figure 3: Approximated V ar 99.5% as a function of N (number of sectors) for an average quality portfolio, characterized by seven rating classes and M = 300 obligors (HHI = 0.0076). sector. Therefore, also the effects of contagion risk my appear somehow diluted, as the number of factors N increases. V ar as a function of the portfolio quality We conclude by analyzing the effects of rating on the value at risk. We consider M = 300 obligors, N = 6 systematic and contagion risk factors, and different values of q. We refer to Fig. 1 for the quality properties of the portfolios, in terms of seven rating classes. We collect the results in Tables 3-5. High Qlty V ar 99.5% Y C C GA q=99% 0.0099 0.005 0.00093 0.0037 0.0013 q=99.5% 0.0114 0.0055 0.00098 0.0041 0.0015 q=99.9% 0.0156 0.0068 0.001 0.005 0.0018 Table 3: Results for a high quality portfolio (25% of BB and below), characterized by seven rating classes, M = 300 obligors (HHI = 0.0076). Ave Qlty V ar 99.5% Y C C GA q=99% 0.021 0.0078 0.0015 0.0055 0.0023 q=99.5% 0.0238 0.0085 0.0016 0.006 0.0025 q=99.9% 0.0307 0.0103 0.0017 0.0075 0.0029 Table 4: Results for an average quality portfolio (50% of BB and below), characterized by seven rating classes, M = 300 obligors (HHI = 0.0076). 11

Low Qlty V ar 99.5% Y C C GA q=99% 0.0324 0.0098 0.0027 0.0061 0.0037 q=99.5% 0.0363 0.0108 0.0029 0.0068 0.004 q=99.9% 0.0453 0.0131 0.0031 0.0088 0.0044 Table 5: Results for a low quality portfolio (79% of BB and below), characterized by seven rating classes, M = 300 obligors (HHI = 0.0076). The results are consistent with the fact that, upon increasing the quality of portfolio loans, V ar q decreases as well as the correction with respect to the asymptotic value at risk. 4.2 Comparative analysis with Monte Carlo simulations We conclude the numerical analysis by comparing our results with those obtained through Monte Carlo simulations. In particular, we refer to the works by Carey (2001) [17] and Gordy (2000) [16]. The former offers a very detailed description of the simulation and the underlying portfolio, the latter presents a comparative analysis between the the CreditMetrics [3] and CreditRisk+ [18] models. Before entering the details of the discussion, we must stress that the works we refer to use an approach which is not based on the decomposition of the value at risk into its asymptotic part and corrections (they actually appeared before the huge literature on this topic). Their methodological framework relies on the simulation of the true loss distribution, through generation of numerous scenarios. However, our model, by construction, is flexible enough to allow for a satisfactory correspondence with both [17] and [16]. Carey 2001 Carey assumes a portfolio with the following features (see Table 1 in [17]): default-mode credit model, flexible number of obligors, but close to 500, maximum loan to one borrower limit in the exposure of about 3%, portfolio quality as expressed in Table 6 (following Moody s criteria), Rating >A 20% Baa 30% Ba 35% B 15% Table 6: Portfolio decomposition into rating classes used by [17]. LGD characterized by mean value µ = 37%, constant across rating classes. 12

Setting up a comparable (though not perfectly matched) portfolio, fixing M = 500 and letting the confidence level q vary, we obtain the following results for the value at risk: q = 95% q = 99% q = 99.5% q = 99.9% Carey 0.0187 0.0271 0.0304 0.0387 Modello 0.0153 0.0229 0.0263 0.0343 Table 7: Comparison between the analytical results of our model and the outcomes of the Monte Carlo simulation, performed by Carey [17] Despite the impossibility to reproduce exactly the portfolio used by Carey, the two models produce values of the V ar which agree on the second digit, the discrepancy being of about 30 40 basis points. Gordy 2000 We now compare our model with those analyzed by Gordy [16], namely the so called restricted CreditMetrics (CM2S), which only accounts for default events, without considering migrations across rating classes, a version of CreditRisk+ (CR+) [18], characterized by a single systematic risk factor, distributed according to a gamma distribution, with unitary mean and standard deviation σ. We stress that the parameter σ can be chosen in an arbitrary way, affecting the final value of V ar to a certain degree. Gordy considers portfolios of M = 5000 loans, of different credit quality, according to the distributions into rating classes proposed in Fig. 1. The mean value of the loss given default is taken equal to µ i = 30%. Systematic factor loadings r i are set to particular values, collected in Table 2, [16]. The numerical analysis is then performed for two different scenarios: the homogeneous portfolio case and in the presence of imperfect granularity. We can reproduce quite accurately the data of the homogeneous case. We assume a portfolio which match closely that by Gordy, except for the number of obligors (we fix M = 400). The results are summarized in Table 8. CR+ Model CM2S σ = 1.5 q = 99.97% 0.0264 0.02649 0.03187 q = 99.5% 0.0174 0.01747 0.02009 q = 99% 0.0153 0.01527 0.01728 Table 8: Comparison between the V ar of our model and those obtained by Gordy [16], for a homogeneous portfolio. The agreement, especially with the CM2S model, obtained through Monte Carlo simulation, is excellent. 13

We conclude, by comparing the results for a portfolio with granularity. Gordy proposes a calibration of the exposures, based on rating [16]. We limit ourselves to keep our specification of granularity, previously exposed (for M = 400 obligors), and collect the results in Tables 9-11: CR+ CR+ CR+ Model CM2S σ = 1 σ = 1.5 σ = 4 q = 99.97% 0.0341 0.02714 0.02736 0.03225 0.05149 q = 99.5% 0.0226 0.01795 0.01847 0.02033 0.02488 q = 99% 0.0199 0.01578 0.01628 0.01749 0.01916 Table 9: Comparison between the V ar of our model and those obtained by Gordy [16], for an average quality portfolio with granularity. CR+ CR+ CR+ Model CM2S σ = 1 σ = 1.5 σ = 4 q = 99.97% 0.0462 0.04558 0.04877 0.05770 0.09251 q = 99.5% 0.0323 0.03124 0.03320 0.03664 0.04504 q = 99% 0.0288 0.02782 0.02936 0.03161 0.03481 Table 10: Comparison between the V ar of our model and those obtained by Gordy [16], for a poor quality portfolio with granularity. CR+ CR+ CR+ Model CM2S σ = 1 σ = 1.5 σ = 4 q = 99.97% 0.0198 0.01342 0.01277 0.01490 0.02345 q = 99.5% 0.0125 0.00847 0.00850 0.00928 0.0121 q = 99% 0.0108 0.00733 0.00745 0.00794 0.00858 Table 11: Comparison between the V ar of our model and those obtained by Gordy [16], for a high quality portfolio with granularity. Despite the fact that our model and those analyzed by Gordy are specified in a different way, we obtain values of the V ar which are compatible, reaching a satisfactory agreement in the case of a low quality portfolio. 5 Conclusions In this paper we have shown how to compute analytically the value at risk for a portfolio of loans, non homogeneous in the exposures and in the presence of both multiple industry-geographic sectors and contagion risk. The key idea consists in approximating the true V ar as a sum of terms: the first contribution is the asymptotic V ar, pertaining to the limiting case of a singlefactor homogeneous portfolio (ASRF), the remaining terms are the corrections due 14

to granularity and the multi-factor set up. Contagion risk affects the adjustments, but do not have any impact on the asymptotic component of value at risk. An important aspect of the model proposed is that it allows to obtain good estimates of the value at risk, without relying on time consuming Monte Carlo simulations. Appendix A Contagion parameters The parameters to be estimated from market data are the factor loadings {g i } and the coefficients {γ il } which appear in the expansion of the composite contagion factor Γ i in terms of the latent variables C l. The idea we propose in our model specification, in order to choose such parameters, is to rely on the information encoded into the revenues generated by single obligors. We assume that data about the revenues of each obligor, R i, are available. In particular, we assume it is possible to quantify the amount of revenues earned from transactions with the infecting segment of each sector. Let us call this quantity R I ik, where k = 1,...,N specifies the sector and i = 1,...,M the single obligor. The coefficient γ ik can be expressed in terms of the revenues data as follows γ ik = C RI ik R i, where the proportionality constant is set to the value C = 1 N k=1 ( ). R I 2 ik R i In this framework, we assume the factor loading g i to be a discretionary parameter, which measures the overall sensitivity of obligor i to contagion risk. B Derivatives of l(y) and ν(y), eq. (12) In order to calculate t q according to eq. (12), we also need explicit expressions for the derivatives of l(y) and ν(y). Given eq.s (10), (11), (14) and (15), the derivatives read M l (y) = w i µ i ˆp i(y), ˆp i(y) = l (y) = a i i=1 M i=1 1 a 2 i n 15 w i µ i ˆp i (y), N 1 (p i ) a i y 1 a 2 i,

ˆp a2 i i (y) = 1 a 2 i N 1 (p i ) a i y 1 a 2 i n N 1 (p i ) a i y 1 a 2 i, ν (y) = 2 ν GA(y) = M i=1 j=1 M w i w j µ i µ j ˆp i(y) N M wi 2 ˆp i(y) i=1 µ 2 i 1 2N N 1 [ˆp j (y)] ρ Y ij N 1 [ˆp i (y)] 1 (ρ Y ij )2 N 1 [ˆp i (y)] ρ Y ii N 1 [ˆp i (y)] 1 (ρ Y ii )2 ˆp j (y), + σi 2. In the presence of credit contagion, all of the above formulas are easily generalized by replacing ρ Y +C ij with ρy ij. References [1] Vasicek O (1991) Limiting loan loss probability distribution, KMV Corporation. [2] Basel Commettee on Banking Supervision, Working Paper 15, Studies on credit risk concentration, November 2006. [3] Gupton G, Finger C and Bhatia M (1997) CreditMetrics-Technical document, April. [4] PortfolioManager, Bohn J, Kealhofer S, Portfolio Management of Default Risk, (2001), working paper, KMV corporation. [5] CreditPortfolioView, McKinsey 1997. [6] Gourieroux C, Laurent J-P and Scaillet O (2000), Sensitivity analysis of values at risk, Journal of Empirical Finance, Vol 7, pp 225-245. [7] Wilde T (2001), Probing granularity, Risk Magazine, August, pp 103-106. [8] Martin R and Wilde T (2002), Unsystematic credit risk, Risk Magazine, November, pp 123-128. [9] Gordy, M B (2003) A risk-factor model foundation for ratings-based bank capital rules, Journal of Financial Intermediation, vol 12, pp 199-232. [10] Emmer S and Tasche D (2003) Calculating credit risk capital charges with the one-factor model, Working paper, September. [11] Pykhtin M (2004) Multi-factor adjustment, Risk Magazine, March, pp 85-90. [12] Davis M and Lo V (2001) Infectious defaults, Quantitative Finance 1, pp 382-387. [13] Egloff D, Leippold M and Vanini P (2004) A simple modelof credit contagion, Working Paper, Universuty of Zurich. 16

[14] Rösch D and Winterfeldt B (2008) Estimating Credit Contagion in a Standard Factor Model, Risk Magazine, August, pp 78-82. [15] Yun-Hoan Oh (2007) Credit concentration risk: extended multi-factor adjustment IRB model. [16] Gordy, M B (2000) A Comparative Anatomy of Credit Risk Models, Journal of Banking and Finance, vol 24, January, pp 119-149. [17] Carey M, (2000) Dimensions of Credit Risk and Their Relationship to Economic Capital Requirements, working paper. [18] CreditRisk+ (Credit Suisse) 17